[seqfan] Re: help needed with sequences related to Kaprekar map

[Richard Mathar]

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For A151962 I get

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4,

    5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3,

    5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6,

    2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2,

    2, 5, 4, 3, 1, 2, 3, 4, 5, 2, 1, 2, 5, 4, 3, 1, 2, 3, 4, 5

with the Maple program

A151949 := proc(n)
        local tup;
        tup := sort(convert(n,base,10)) ;
        add( (op(i,tup)-op(-i,tup)) *10^(i-1),i=1..nops(tup)) :
end:

A151962 := proc(n)
        local tra,x ;
        tra := [n] ;
        x := n ;
        while true do
                x := A151949(x) ;
                if member(x,tra,'l') then
                        RETURN(l-1) ;
                fi;
                tra := [op(tra),x] :
        od:
end:

seq(A151962(n),n=0..120) ;


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and for A151963 we have

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5,

    6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4,

    6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7,

    3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3,

    3, 6, 5, 4, 2, 3, 4, 5, 6, 3, 2, 3, 6, 5, 4, 2, 3, 4, 5, 6

by the Maple program

A151949 := proc(n)
        local tup;
        tup := sort(convert(n,base,10)) ;
        add( (op(i,tup)-op(-i,tup)) *10^(i-1),i=1..nops(tup)) :
end:

A151963 := proc(n)
        local tra,x ;
        tra := [n] ;
        x := n ;
        while true do
                x := A151949(x) ;
                if x in tra then
                        RETURN(nops(tra)) ;
                fi;
                tra := [op(tra),x] :
        od:
end:

seq(A151963(n),n=0..120) ;

---
For A151957 I get

0, 495, 6174, 62964, 420876, 7509843, 64308654, 753098643, 6431088654,

    86420987532, 643330866654, 8764209875322, 64333308666654, 885432098765412,

    6543331088666544, 88543320987665412, 975533110888664421,

    8854333209876665412, 65554333108866654444, 987654321098876543211,

    8876644220998775533212, 98765433210988766543211, 643333333330866666666654,

    9876555432109887654443211, 64333333331108886666666654,

    987655554321098876544443211, 6433333333311088866666666654,

    98765543332110988876665443211, 988766544322109988776554332111,

    8886664443222099987776555333112, 98876654433322099877666554332111,

    987654333321111098888876666543211, 6433111111111111088888888888886654,

    98765433321111110988888887666543211, 988766544322111109988888776554332111,

    9876554332111111109888888887665443211,

    98876655554432211099888776554444332111,

with the program

A151949 := proc(n)
        local tup;
        tup := sort(convert(n,base,10)) ;
        add( (op(i,tup)-op(-i,tup)) *10^(i-1),i=1..nops(tup)) :
end:

A151957 := proc(n)
        local tra,x ;
        x := 10^(n-1)+2 ;
        tra := [x] ;
        while true do
                x := A151949(x) ;
                if member(x,tra,'l') then
                        op(l..nops(tra),tra) ;
                        RETURN(min(%)) ;
                fi;
                tra := [op(tra),x] :
        od:
end:

seq(A151957(n),n=2..60) ;
---------
Richard J. Mathar